Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/257

and we have for the value of $X_{1}$,

$X_{1}=i_{2}{\frac {d}{dx_{1}}}\int {\left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)ds}.$

(3)

Now let us suppose that the displacement consists in moving every point of the circuit through a distance δx in the direction of x, δx being any continuous function of s, so that the different parts of the circuit move independently of each other, while the circuit remains continuous and closed.

Also let X be the total force in the direction of x acting on the part of the circuit from s = 0 to s = s, then the part corresponding to the element ds will be ${\frac {dX}{ds}}ds$. We shall then have the following expression for the work done by the force during the displacement,

$\int {{\frac {dX}{ds}}\delta x\,ds}=i_{2}\int {{\frac {d}{d\delta x}}\left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)\delta x\,ds}$

(4)

where the integration is to be extended round the closed curve, remembering that δx is an arbitrary function of s. We may there fore perform the differentiation with respect to δx in the same way that we differentiated with respect to t in Art. 598, remembering that

${\frac {dx}{d\delta x}}=1,\quad {\frac {dy}{d\delta x}}=0\quad {\text{and }}{\frac {dz}{d\delta x}}=0.$

(5)

We thus find

$\int {{\frac {dX}{ds}}\delta x\,ds}=i_{2}\int {\left(c{\frac {dy}{ds}}-b{\frac {dz}{ds}}\right)\delta x\,ds}+\int {{\frac {d}{ds}}(F\delta x)ds}.$

(6)

The last term vanishes when the integration is extended round the closed curve, and since the equation must hold for all forms of the function δx, we must have

${\frac {dX}{ds}}=i_{2}\left(c{\frac {dy}{ds}}-b{\frac {dz}{ds}}\right)$

(7)

an equation which gives the force parallel to x on any element of the circuit. The forces parallel to y and z are

${\frac {dY}{ds}}=i_{2}\left(a{\frac {dz}{ds}}-c{\frac {dx}{ds}}\right)$

(8)

${\frac {dZ}{ds}}=i_{2}\left(b{\frac {dx}{ds}}-a{\frac {dy}{ds}}\right)$

(9)

The resultant force on the element is given in direction and magnitude by the quaternion expression $i_{2}Vd\rho {\mathfrak {B}}$, where i₂ is the numerical measure of the current, and dρ and ${\mathfrak {B}}$ are vectors